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\usepackage{amsmath,amssymb,amsthm} % 数学公式与定理环境
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\title{常微分方程考试A}
\author{2022级数学与应用数学1班}
\date{2023年秋季}

\begin{document}

\maketitle


本次考试共10题，每题10分。

\begin{enumerate}\itemsep1em

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\item %1
设 $a,b,c$ 是实数常数。判断微分方程 $(2ax+by)dx + (bx+2cy)dy=0$ 是否为恰当方程，并求解。

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\item %2
设连续函数 $f(x)$ 在区间 $(-\infty,\infty)$ 上是有界的。求出微分方程 $\frac{dy}{dx} + y = f(x)$ 的通解。
有多少个解在区间 $(-\infty,\infty)$ 上是有界的？

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\item %3
记 $p=\frac{dy}{dx}$, 求隐式微分方程 $y=(x+2)p+p^2$ 的奇解。 

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\item %4
设有一根长度为 $\ell$ 的不计质量的细线，上端固定，下端悬挂一个质量为 $m$ 的小球，组成一个单摆。设空气阻力与速度成正比，根据牛顿第二运动定律，导出单摆的运动方程。

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\item %5
求下述齐次线性微分方程组的通解：
\begin{eqnarray*}
\frac{d}{dt}
\begin{bmatrix} x \\ y \\ z \end{bmatrix}
=
\begin{bmatrix} 2&4&0 \\ 0&2&0 \\ 0&0&3 \end{bmatrix}
\begin{bmatrix} x \\ y \\ z \end{bmatrix}.
\end{eqnarray*}

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\item %6
求常系数线性微分方程的通解：$y''+3y'+2y=4x$. 

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\item %7
求初值问题的幂级数解的前五项：
$%\begin{eqnarray*}
y''+xy'+y=0, \,\, y(0)=1, y'(0)=1. 
$%\end{eqnarray*}

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\item %8
按定义验证 $x=0$ 是微分方程 $xy''-y=0$ 的正则奇点，并求出广义幂级数解的前四项。

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\item %9
使用李雅普诺夫方法判断下述方程的零解的稳定性：
\begin{eqnarray*}
\left\{ \begin{array}{rcl}
\frac{dx}{dt} &=& y-xy^4, \\
\frac{dy}{dt} &=& -x-x^2y.
\end{array}\right.
\end{eqnarray*}

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\item %10
求下列方程的奇点，判断奇点的类型，并作出奇点附近的相图：
\begin{eqnarray*}
\left\{ \begin{array}{rcl}
\frac{dx}{dt} &=& 5x-2y, \\
\frac{dy}{dt} &=& 6x-2y. 
\end{array}\right.
\end{eqnarray*}

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\end{enumerate}

\end{document}
